GATE 2023 PH – Question 63
Consider 6 identical, non-interacting, spin 1 2 atoms arranged on a crystal lattice at absolute temperature T. The z-component of the magnetic moment of each of these atoms can be ± μB. If P and Q are the probabilities of the net magnetic moment of the solid being 2μB and 6μB respectively, what is the value of P Q (in integer)?
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Correct answer: 15
Explanation
There are 6 identical, non-interacting spin-½ atoms on lattice sites. Each atom has a magnetic moment $+\mu_B$ (spin up) or $-\mu_B$ (spin down). The atoms sit on different sites, so they are distinguishable by position. At any temperature, each configuration (microstate) of a given total energy has the same probability. (In zero field, all the configurations are equally likely.)
If $n$ atoms are up, the net moment is
$$M=n\mu_B-(6-n)\mu_B=(2n-6)\mu_B .$$
- **$M=2\mu_B$:** $2n-6=2$, so $n=4$. The number of microstates is $\binom64=15$.
- **$M=6\mu_B$:** $2n-6=6$, so $n=6$ (all up). The number of microstates is $\binom66=1$.
$$\frac PQ=\frac{15}{1}=\mathbf{15}.$$